# Ihara’s theorem

Let $\Gamma$ be a discrete, torsion-free subgroup of $\mathrm{SL}_{2}\mathbb{Q}_{p}$ (where $\mathbb{Q}_{p}$ is the field of $p$-adic numbers (http://planetmath.org/PAdicIntegers)). Then $\Gamma$ is free.

###### Proof, or a sketch thereof.

There exists a $p+1$ regular tree $X$ on which $\mathrm{SL}_{2}\mathbb{Q}_{p}$ acts, with stabilizer $\mathrm{SL}_{2}\mathbb{Z}_{p}$ (here, $\mathbb{Z}_{p}$ denotes the ring of $p$-adic integers (http://planetmath.org/PAdicIntegers)). Since $\mathbb{Z}_{p}$ is compact in its profinite topology, so is $\mathrm{SL}_{2}\mathbb{Z}_{p}$. Thus, $\mathrm{SL}_{2}\mathbb{Z}_{p}\cap\Gamma$ must be compact, discrete and torsion-free. Since compact and discrete implies finite, the only such group is trivial. Thus, $\Gamma$ acts freely on $X$. Since groups acting freely on trees are free, $\Gamma$ is free. ∎

Title Ihara’s theorem IharasTheorem 2013-03-22 13:54:26 2013-03-22 13:54:26 bwebste (988) bwebste (988) 9 bwebste (988) Theorem msc 20G25